AN OPTIMAL FILTERING METHOD FOR THE SIDEWAYS HEAT-EQUATION

The authors consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=x and they want to determine a solution for 0<x<x. This problem, which can be called the sideways heat equation, is ill-posed: the solution (if it exists) does not depend continuously on the data. Continuous dependence is restored if a bound is imposed on the solution at x=0, and Holder-type error estimates can be obtained for this stabilised problem. They consider a filtering method (defined using the Fourier transform) for computing an approximate solution of the sideways heat equation.